HungDo_template.utf8

Outlines

Equations
- Maxwell Equations
- Schrödinger Equation
Interactive Plots
Tables
Interactive Maps
R Shiny Apps #1: Simple Nonlinear Exponential Fitting
R Shiny Apps #2: Predict next Word using a simple n-gram Language Model
Embedded Websites

Maxwell Equations

The coupled partial differential equations that form the foundation of classical electromagnetism. The integral (left) and differential (right) form.

\begin{array}{ll} \int \int ◯ (\vec{D} \cdot \vec{n}) d^{2} A = Q_{f r e e, i n c l u d e d} & \nabla \cdot \vec{D} = ρ_{f r e e} \\ \int \int ◯ (\vec{B} \cdot \vec{n}) d^{2} A = 0 & \nabla \cdot \vec{B} = 0 \\ \oint \vec{E} \cdot d \vec{s} = - \frac{d Φ}{d t} & \nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t} \\ \oint \vec{H} \cdot d \vec{s} = I_{f r e e, i n c l u d e d} + \frac{d Ψ}{d t} & \nabla \times \vec{H} = {\vec{J}}_{f r e e} + \frac{\partial \vec{D}}{\partial t} \end{array}

$\def\oiint{\int\hspace{-2ex}\int\hspace{-3ex}\bigcirc~} \def\Q#1#2{\frac{\partial #1}{\partial #2}} \begin{array}{ll} \displaystyle\oiint(\vec{D}\cdot\vec{n}\,)d^2A=Q_{\rm free,included}~~~~~~~~~~~~~ &\displaystyle\nabla\cdot\vec{D}=\rho_{\rm free}\\ \displaystyle\oiint(\vec{B}\cdot\vec{n}\,)d^2A=0 &\displaystyle\nabla\cdot\vec{B}=0\\ \displaystyle\oint\vec{E}\cdot d\vec{s}=-\frac{d\Phi}{dt} &\displaystyle\nabla\times\vec{E}=-\Q{\vec{B}}{t}\\ \displaystyle\oint\vec{H}\cdot d\vec{s}=I_{\rm free,included}+\frac{d\Psi}{dt} &\displaystyle\nabla\times\vec{H}=\vec{J}_{\rm free}+\Q{\vec{D}}{t} \end{array}$

Schrödinger Equation

The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system.

- \frac{ℏ^{2}}{2 m} \nabla^{2} ψ + U ψ = E ψ = i ℏ \frac{\partial ψ}{\partial t}

$\displaystyle -\frac{\hbar^2}{2m}\nabla^2\psi+U\psi=E\psi=i\hbar\Q{\psi}{t}$

Interactive Plots

Tables

Let’s use a linear model to fit miles per gallon (mpg) as a function of weight (wt) and the number of cylinders (cyl).

library(pander)
data("mtcars")
fit <- lm(mpg~wt+cyl, data=mtcars)
pander(fit, type = 'html', title = 'Linear Model Outputs')

Fitting linear model: mpg ~ wt + cyl
	Estimate	Std. Error	t value	Pr(>\|t\|)
(Intercept)	39.69	1.715	23.14	3.043e-20
wt	-3.191	0.7569	-4.216	0.000222
cyl	-1.508	0.4147	-3.636	0.001064

Interactive Maps

R Shiny Apps #1: Simple Nonlinear Exponential Fitting

Here is the weblink to a very simple R shiny app that performs the nonlinear exponential fitting.
Here is the weblink to the slide deck that describes the app.

R Shiny Apps #2: Predicting Next Word

Here is the weblink to a very simple R shiny app that predict the next word using a simple n-gram language model.

Hung Do’s Homepage (weblink)

Rmarkdown Presentation Template

Hung P. Do, PhD, MSEE